Proposition 8

Let B be a simple subalgebra of A. Denote the center of B by L and the commutant of B in A by $B'$. Then $B'$ is a central simple algebra over the field L; we denote its reduced degree by $r$. For every element $b$ of B, we have the relations

(56) Pcrd$_{A/K}(b; X) = N_{L[X]/K[X]}$(Pcrd$_{B/L}(b; X)$)$^r$,

(57) Trd$_{A/K}(b) =r$ Tr$_{L/K}$(Trd$_{B/K}(b)$),

(58) Nrd$_{A/K}(b) = N_{L/K}$(Nrd$_{B/L}(b)$)$^r$.